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Properties

Label 12.12.2736443806...4896.1
Degree $12$
Signature $[12, 0]$
Discriminant $2^{32}\cdot 3^{12}\cdot 337^{4}\cdot 310501^{4}$
Root discriminant $8976.33$
Ramified primes $2, 3, 337, 310501$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $M_{11}$ (as 12T272)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5270752514492878, -3666673786739256, -401077519158720, 32827558924192, 3652512071076, -66045530544, -9806288196, 48056976, 11072751, -11000, -5526, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 5526*x^10 - 11000*x^9 + 11072751*x^8 + 48056976*x^7 - 9806288196*x^6 - 66045530544*x^5 + 3652512071076*x^4 + 32827558924192*x^3 - 401077519158720*x^2 - 3666673786739256*x + 5270752514492878)
 
gp: K = bnfinit(x^12 - 5526*x^10 - 11000*x^9 + 11072751*x^8 + 48056976*x^7 - 9806288196*x^6 - 66045530544*x^5 + 3652512071076*x^4 + 32827558924192*x^3 - 401077519158720*x^2 - 3666673786739256*x + 5270752514492878, 1)
 

Normalized defining polynomial

\( x^{12} - 5526 x^{10} - 11000 x^{9} + 11072751 x^{8} + 48056976 x^{7} - 9806288196 x^{6} - 66045530544 x^{5} + 3652512071076 x^{4} + 32827558924192 x^{3} - 401077519158720 x^{2} - 3666673786739256 x + 5270752514492878 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(273644380620093450686292199131386889530712784896=2^{32}\cdot 3^{12}\cdot 337^{4}\cdot 310501^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $8976.33$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 337, 310501$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} + \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{985678598339832661686396681933955284128914571457240756903869792839956} a^{11} + \frac{48547815486620616490534752127952089899264681575794355204143484093141}{985678598339832661686396681933955284128914571457240756903869792839956} a^{10} - \frac{158197683913685928886366839914886347833055975509469244835512666450461}{985678598339832661686396681933955284128914571457240756903869792839956} a^{9} - \frac{39119729772956975868247761329444792356941911280230725682348567215231}{328559532779944220562132227311318428042971523819080252301289930946652} a^{8} - \frac{337291335351412568834777146693520577613204529256432405415260399841}{54759922129990703427022037885219738007161920636513375383548321824442} a^{7} - \frac{18904574850590731397596387724436318623618328249068080677397070120423}{164279766389972110281066113655659214021485761909540126150644965473326} a^{6} + \frac{32738204992985751756777082056781356628541917949903941352337684829967}{164279766389972110281066113655659214021485761909540126150644965473326} a^{5} + \frac{20946163850828709792979101686346060616956223784702938012663578877385}{54759922129990703427022037885219738007161920636513375383548321824442} a^{4} + \frac{52265638186905312914472363277110914149625158613384908143469835567951}{164279766389972110281066113655659214021485761909540126150644965473326} a^{3} - \frac{205875409927210967727491299370073839410419200014553142026319203400661}{492839299169916330843198340966977642064457285728620378451934896419978} a^{2} - \frac{11305467892337746861185163829695189629201080900729575979847787195049}{492839299169916330843198340966977642064457285728620378451934896419978} a + \frac{109361131686897700677171175417287015437851209029374318010448144261479}{492839299169916330843198340966977642064457285728620378451934896419978}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 110704913560000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$M_{11}$ (as 12T272):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7920
The 10 conjugacy class representatives for $M_{11}$
Character table for $M_{11}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 11 sibling: data not computed
Degree 22 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.11.0.1}{11} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.11.0.1}{11} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.11.0.1}{11} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.4.8.5$x^{4} + 2 x^{2} + 4 x + 6$$4$$1$$8$$D_{4}$$[2, 3]^{2}$
2.8.24.66$x^{8} + 20 x^{4} + 52$$8$$1$$24$$QD_{16}$$[2, 3, 4]^{2}$
$3$3.12.12.27$x^{12} - 12 x^{11} + 3 x^{10} - 9 x^{9} + 3 x^{8} + 3 x^{7} + 6 x^{6} + 9 x^{3} + 9 x + 9$$6$$2$$12$12T47$[5/4, 5/4]_{4}^{2}$
337Data not computed
310501Data not computed